Theorem grpideu 18892

Description: The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)

Hypotheses

Ref	Expression
grpcl.b	⊢ 𝐵 = (Base‘𝐺)
grpcl.p	⊢ + = (+g‘𝐺)
grpinvex.p	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
grpideu	⊢ (𝐺 ∈ Grp → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))

Distinct variable groups:   𝑥,𝑢,𝐵   𝑢,𝐺,𝑥   𝑢, + ,𝑥   𝑥, 0

Allowed substitution hint:   0 (𝑢)

Proof of Theorem grpideu

Step	Hyp	Ref	Expression
1		grpmnd 18888	. 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
2		grpcl.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
3		grpcl.p	. . 3 ⊢ + = (+g‘𝐺)
4	2, 3	mndideu 18696	. 2 ⊢ (𝐺 ∈ Mnd → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
5	1, 4	syl 17	1 ⊢ (𝐺 ∈ Grp → ∃!𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056  ∃!wreu 3369  ‘cfv 6542  (class class class)co 7414  Basecbs 17171  +_gcplusg 17224  0_gc0g 17412  Mndcmnd 18685  Grpcgrp 18881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-grp 18884

This theorem is referenced by: (None)